3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 333 theorem to show that there exist two positive measures a, /? on X such that X2(u) — c^iO) = J u doc. and xiO) - ^(w) = I u d^ Deduce that if x is a character of B, then the Gleason part (Problem 18(b)) containing x is the set of characters for which a representative measure has base ft, for at least one representative measure /* for X- (c) Suppose x is a character for which the representative measure is unique. If #' belongs to the Gleason part containing ^, show that the representative measure /z' for x' is also unique, and that - (d) Suppose that there exists a hyperextremal representative measure ft for x (Problem 11). Show that if a character x' admits a representative measure ft' with base jit, then ft' is the only representative measure for #' with base ft. (If ft'=/- ft and /x" = g • ft are two representative measures for ^', show that /— g is zero almost everywhere with respect to ft, by using Problem 13(j).) 20. Let A and B be two commutative Banach algebras having unit elements and h : A -* B a homomorphism which sends unit element to unit element. (a) Let F be the graph of h in A x B. For each character x of B, show that the mapping (#, y) t—>• %(/*(#)) — x(y) is continuous on A X B. Deduce that if (a, b) is in the closure of F, then x(h(<*)) = xW- (b) Deduce from (a) that if B is without radical (Section 15.2, Problem 7) then h is necessarily continuous (use the closed-graph theorem). In particular, for a commuta- tive C-algebra having a unit element, in which the intersection of the maximal ideals is {0}, two norms which define Banach algebra structures are necessarily equivalent. 21. Let 2 be the algebra of indefinitely differentiable complex-valued functions on [0, 1]. (a) Let A be a subalgebra of ^, containing the unit element and endowed with a norm which makes A a Banach algebra. Show that there exists a sequence (wn)n>0 of finite real numbers ^0 such that, for all functions x e A, we have sup |x<">(0| = 0(mn) O^r^l as n increases indefinitely. (If @n is the Banach algebra of Section 15.1, Problem 1, then the canonical injection of A into ^n is continuous (Problem 20).) (b) Show that there exists no norm on 2 with respect to which B is a Banach algebra. (Use (a) and Problem 4 of Section 8.14.) 22. Let A be a separable Banach algebra with unit element. Show that if a ^= A is a left ideal of A, there exists a maximal left ideal containing a. (Observe that if n is a closed left ideal and (xm) a dense sequence in A such that either xm e n orn + Axm = A for each m, then n is maximal.) Show that the radical of A (Section 15.2, Problem 7) is the intersection of the maximal left ideals of A, and is also the intersection of the maximal right ideals of A.a) denote the least upper bound of